---
title: "Velocity for fully turbulent flow in rough pipe"
subject: "Hydraulics"
chapter: 1
key: "semester_4/hydraulics:fully-turbulent-velocity-1200mm-pipe"
source: https://www.bctengineer.com/q/semester_4/hydraulics/fully-turbulent-velocity-1200mm-pipe
site: BCT Engineering
---

# Velocity for fully turbulent flow in rough pipe

Hydraulics · Chapter 1: Pipe flow

- Asked in: 2080 Chaitra, 4 marks
- Video: [Fully-rough turbulent flows in pipes & man-made channels – Part 1](https://www.youtube.com/watch?v=plov1RL_QMw) — Hubert Chanson
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/fully-turbulent-velocity-1200mm-pipe
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-1

## Question

A 120 cm diameter pipe has equivalent roughness height of 1 mm. What will be the velocity at which the roughness causes the flow to become fully turbulent?

## Short answer

The fully rough criterion is u₊k/ν ≥ 100. Using ν = 10⁻⁶ m²/s, k = 0.001 m, D = 1.2 m, the rough-pipe friction factor f = 0.0188, then u₊ = 0.1 m/s and V = u₊/√(f/8) ≈ 2.06 m/s.

## Answer

**Criterion.** Flow is fully turbulent (hydrodynamically rough) when the roughness Reynolds number (Nikuradse) is
$$
\frac{u_* k}{\nu} \ge 100
$$
Take $\nu = 1.0\times10^{-6}$ m²/s (water at about 20 °C), $k = 1$ mm $= 0.001$ m, $D = 1.2$ m.

**Step 1: rough-pipe friction factor** (von Karman-Prandtl, $D/k = 1200$):
$$
\frac{1}{\sqrt f} = 2\log_{10}\frac{D}{k} + 1.14 \;\Rightarrow\; f = 0.01877, \qquad \sqrt{f/8} = 0.04844
$$

**Step 2: shear velocity at the limit**
$$
u_* = \frac{100\,\nu}{k} = \frac{100(1.0\times10^{-6})}{0.001} = 0.100\ \text{m/s}
$$

**Step 3: mean velocity**
$$
V = \frac{u_*}{\sqrt{f/8}} = \frac{0.100}{0.04844} = 2.06\ \text{m/s}
$$

**Answer:** The flow becomes fully turbulent (rough) for $V \ge 2.06$ m/s, about 2 m/s. (If the limit $u_*k/\nu = 70$ is used, the velocity is 1.45 m/s.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What roughness Reynolds number marks fully turbulent (rough) flow?
   A. u* k / nu >= 100
   B. u* k / nu <= 4
   C. Re >= 2000
   D. u* k / nu = 1
   Answer: A. Nikuradse's limit for hydrodynamically rough flow is u*k/nu >= 100. (Key point: Rough criterion)

2. What is u* at the limit for k = 1 mm and nu = 1e-6 m^2/s?
   A. 0.01 m/s
   B. 1.0 m/s
   C. 0.1 m/s
   D. 0.0001 m/s
   Answer: C. u* = 100 nu / k = 100(1e-6)/0.001 = 0.1 m/s. (Key point: Shear velocity)

3. How is mean velocity obtained from u*?
   A. V = u* sqrt(f/8)
   B. V = u* / sqrt(f/8), with f from the rough-pipe law
   C. V = 8 u*
   D. V = u* f
   Answer: B. u* = V sqrt(f/8), so V = 0.1/0.04844 = 2.06 m/s. (Key point: Mean velocity)

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